It suffices to show that \(\varphi ({T})\)  is a thin set in \(\mathbb {P}^2(\mathbb {Q})\) . By abuse of notation we view \({T}\)  as a subset of \(\mathbb {P}^3(\mathbb {Q})\)  via the map \((z_1,\ldots , z_4) \mapsto [z_1:\cdots : z_4]\) . We begin by showing that \({T}\)  is a thin subset \(\mathbb {P}^3(\mathbb {Q})\) . Consider the weighted projective space \(\mathbb {P}_{\mathbb {Q}}(2,1,1,1,1)\)  with variables \(t, z_1, \ldots , z_4\)  (we refer the reader to [1]} for the basic definitions pertaining to weighted projective spaces). We have an embedding
\(\nu :\mathbb {P}(2,1,1,1,1) &\hookrightarrow \mathbb {P}^{10}\\[t:z_1:\cdots : z_4] &\mapsto [t: z_1^2:z_1z_2:\cdots :z_4^2],\) 

which on the \(z_i\) -variables is the Veronese embedding of degree 2. The polynomial \(f(t,z_1,\ldots ,z_4) = t^2-z_1z_2z_3z_4\)  is weighted homogeneous of degree 4, and so defines a subvariety \(V\)  of \(\mathbb {P}(2,1,1,1,1)\) . Let \(Y\)  denote the image of \(\nu \)  and write \(t,y_{11},\ldots , y_{44}\)  for variables on \(\mathbb {P}^{10}\) . Then \(\nu (V)\)  is a hypersurface of \(Y\)  defined by the equation \(t^2 = y_{12}y_{34}\) . From this we see that \(V\)  is integral, projective and of dimension 3.
Consider the morphism \(\pi : V \rightarrow \mathbb {P}^3\)  defined by \([t: z_1: \cdots z_4]\rightarrow [z_1: \cdots : z_4]\) . This is étale of degree 2 on the open subset \(V^{\prime }\)  of \(V\)  defined by \(z_1\cdots z_4 \ne 0\) . The set \(W\subseteq \mathbb {P}^3({\mathbb {Q}})\)  defined by the equation \(z_1\cdots z_4=0\)  is a type I thin set, and \({T}=\pi (V^{\prime }(\mathbb {Q}))\cup W\) , so we deduce that \({T}\)  is thin in \(\mathbb {P}^3_{\mathbb {Q}}\) .
We can now show that \(\varphi ({T})\)  is thin in \(\mathbb {P}^2_{\mathbb {Q}}\) . To do this, we intersect \(\mathbb {P}^3,V^{\prime }\)  and \(W\)  with the hyperplane \(H\)  defined by the equation \(z_1+z_2+z_3= z_4\) . The map \(\pi \)  is ramified along the set \(W\) , which is a union of hyperplanes \(H_i\)  given by \(\lbrace z_i=0\rbrace \) . Since the intersection of \(H\)  with the \(H_i\)  is smooth and transversal, it follows from [2]} that \(\pi ((V^{\prime }(\mathbb {Q})\cap H(\mathbb {Q}))\cup (W\cap H(\mathbb {Q}))\)  is thin in \(H(\mathbb {Q})\) . The image of this set under the obvious isomorphism \(H \cong \mathbb {P}^2\)  sending \([z_1:\cdots :z_4]\)  to \([z_1:z_2:z_3]\)  is precisely \(\varphi ({T})\) , so we conclude that \(\varphi ({T})\)  is a thin set in \(\mathbb {P}^2_{\mathbb {Q}}\) .

